Frequency dependence of fracture stiffness

The frequency dependence of fracture specific stiffness, for seismic or ultrasonic wave transmission, is shown to be a simple scaling property of fractures with spatially inhomogeneous distributions of stiffness. Different frequencies sample different subsets of the fracture geometry. Therefore the frequency dependence may be a simple consequence of fracture geometry and requires no additional dynamical mechanism. In this letter, the dynamic fracture stiffness is determined based on the displacement discontinuity model for wave transmission across a fracture. Local transmission coefficients are assumed to depend on the local static stiffness. For a fracture with a single value of static stiffness, the dynamic stiffness is frequency independent. A strongly inhomogeneous distribution of fracture stiffnesses produces a strong frequency dependence for the dynamic stiffness.